Optimal convex shapes for concave functionals

¹ Laboratoire de Mathématiques UMR 5127, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-du-Lac, France
² Dipartimento di Matematica, Politecnico, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy
ilaria.fragala@polimi.it
³ Ceremade UMR 7534, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France

Received: 29 November 2010

Abstract

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetric-like inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Pólya-Szegö problem.